curve #3667

y² + xy + y = x³ − x² − 96024481009434638942x + 362120228655799988685458307309

a-invariants: [1, -1, 1, -96024481009434638942, 362120228655799988685458307309]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 2542465729285305792210
discriminant (Δ): 17815595092280169081648964232294920753115884408704860160000
Faltings height: 10.1806
naive height: 149.6470
primes of bad reduction: 2, 3, 5, 11, 19, 31, 37, 43, 103, 389, 487, 140449
regulator: 137200.343000733322394238313457040513446342359688666981
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:48 UTC
last updated: 2026-09-29 19:29:48 UTC

Witness: 8 independent points log in to add more points →

(-7938235783, 790037265530691)
(5065687457, 75375082920051)
(5543816669, 12695077276587)
(5592123257, 3933317725251)
(27037665089, 4187029708157067)
(34998349689, 6314280344267843)
(-110641817047/49, 258375402074895813/343)
(1486947843993/289, 320849338232201683/4913)

Commentary

Specialization at t = 389/90 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:29:48 UTC · history